3.17.58 \(\int \frac {12+e^{e^x} (2-20 e^{2 x}+e^x (20-2 x))+e^x (80+40 x)}{36+e^{2 e^x}+24 x+4 x^2+e^{e^x} (12+4 x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {20 e^x+2 x}{6+e^{e^x}+2 x} \]

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Rubi [F]  time = 1.74, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {12+e^{e^x} \left (2-20 e^{2 x}+e^x (20-2 x)\right )+e^x (80+40 x)}{36+e^{2 e^x}+24 x+4 x^2+e^{e^x} (12+4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12 + E^E^x*(2 - 20*E^(2*x) + E^x*(20 - 2*x)) + E^x*(80 + 40*x))/(36 + E^(2*E^x) + 24*x + 4*x^2 + E^E^x*(1
2 + 4*x)),x]

[Out]

-40*Defer[Int][E^x/(6 + E^E^x + 2*x)^2, x] - 20*Defer[Int][E^(E^x + 2*x)/(6 + E^E^x + 2*x)^2, x] - 4*Defer[Int
][x/(6 + E^E^x + 2*x)^2, x] + 12*Defer[Int][(E^x*x)/(6 + E^E^x + 2*x)^2, x] + 4*Defer[Int][(E^x*x^2)/(6 + E^E^
x + 2*x)^2, x] + 2*Defer[Int][(6 + E^E^x + 2*x)^(-1), x] + 20*Defer[Int][E^x/(6 + E^E^x + 2*x), x] - 2*Defer[I
nt][(E^x*x)/(6 + E^E^x + 2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12+e^{e^x} \left (2-20 e^{2 x}+e^x (20-2 x)\right )+e^x (80+40 x)}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ &=\int \left (-\frac {20 e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2}+\frac {2 \left (6+e^{e^x}\right )}{\left (6+e^{e^x}+2 x\right )^2}-\frac {2 e^x \left (-40-10 e^{e^x}-20 x+e^{e^x} x\right )}{\left (6+e^{e^x}+2 x\right )^2}\right ) \, dx\\ &=2 \int \frac {6+e^{e^x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx-2 \int \frac {e^x \left (-40-10 e^{e^x}-20 x+e^{e^x} x\right )}{\left (6+e^{e^x}+2 x\right )^2} \, dx-20 \int \frac {e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ &=2 \int \left (-\frac {2 x}{\left (6+e^{e^x}+2 x\right )^2}+\frac {1}{6+e^{e^x}+2 x}\right ) \, dx-2 \int \left (\frac {e^x (-10+x)}{6+e^{e^x}+2 x}-\frac {2 e^x \left (-10+3 x+x^2\right )}{\left (6+e^{e^x}+2 x\right )^2}\right ) \, dx-20 \int \frac {e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ &=2 \int \frac {1}{6+e^{e^x}+2 x} \, dx-2 \int \frac {e^x (-10+x)}{6+e^{e^x}+2 x} \, dx-4 \int \frac {x}{\left (6+e^{e^x}+2 x\right )^2} \, dx+4 \int \frac {e^x \left (-10+3 x+x^2\right )}{\left (6+e^{e^x}+2 x\right )^2} \, dx-20 \int \frac {e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ &=2 \int \frac {1}{6+e^{e^x}+2 x} \, dx-2 \int \left (-\frac {10 e^x}{6+e^{e^x}+2 x}+\frac {e^x x}{6+e^{e^x}+2 x}\right ) \, dx-4 \int \frac {x}{\left (6+e^{e^x}+2 x\right )^2} \, dx+4 \int \left (-\frac {10 e^x}{\left (6+e^{e^x}+2 x\right )^2}+\frac {3 e^x x}{\left (6+e^{e^x}+2 x\right )^2}+\frac {e^x x^2}{\left (6+e^{e^x}+2 x\right )^2}\right ) \, dx-20 \int \frac {e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ &=2 \int \frac {1}{6+e^{e^x}+2 x} \, dx-2 \int \frac {e^x x}{6+e^{e^x}+2 x} \, dx-4 \int \frac {x}{\left (6+e^{e^x}+2 x\right )^2} \, dx+4 \int \frac {e^x x^2}{\left (6+e^{e^x}+2 x\right )^2} \, dx+12 \int \frac {e^x x}{\left (6+e^{e^x}+2 x\right )^2} \, dx-20 \int \frac {e^{e^x+2 x}}{\left (6+e^{e^x}+2 x\right )^2} \, dx+20 \int \frac {e^x}{6+e^{e^x}+2 x} \, dx-40 \int \frac {e^x}{\left (6+e^{e^x}+2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 21, normalized size = 0.95 \begin {gather*} \frac {2 \left (10 e^x+x\right )}{6+e^{e^x}+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + E^E^x*(2 - 20*E^(2*x) + E^x*(20 - 2*x)) + E^x*(80 + 40*x))/(36 + E^(2*E^x) + 24*x + 4*x^2 + E^
E^x*(12 + 4*x)),x]

[Out]

(2*(10*E^x + x))/(6 + E^E^x + 2*x)

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fricas [A]  time = 0.76, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (x + 10 \, e^{x}\right )}}{2 \, x + e^{\left (e^{x}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="fricas")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

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giac [A]  time = 0.24, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (x + 10 \, e^{x}\right )}}{2 \, x + e^{\left (e^{x}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="giac")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

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maple [A]  time = 0.08, size = 19, normalized size = 0.86

method	result	size
risch	\(\frac {2 x +20 \,{\mathrm e}^{x}}{2 x +{\mathrm e}^{{\mathrm e}^{x}}+6}\)	\(19\)
norman	\(\frac {2 x +20 \,{\mathrm e}^{x}}{2 x +{\mathrm e}^{{\mathrm e}^{x}}+6}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(x))+4*
x^2+24*x+36),x,method=_RETURNVERBOSE)

[Out]

2*(10*exp(x)+x)/(2*x+exp(exp(x))+6)

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maxima [A]  time = 0.67, size = 18, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (x + 10 \, e^{x}\right )}}{2 \, x + e^{\left (e^{x}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="maxima")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (40\,x+80\right )-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (20\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (2\,x-20\right )-2\right )+12}{24\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+4\,x^2+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x+12\right )+36} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(40*x + 80) - exp(exp(x))*(20*exp(2*x) + exp(x)*(2*x - 20) - 2) + 12)/(24*x + exp(2*exp(x)) + 4*x^
2 + exp(exp(x))*(4*x + 12) + 36),x)

[Out]

int((exp(x)*(40*x + 80) - exp(exp(x))*(20*exp(2*x) + exp(x)*(2*x - 20) - 2) + 12)/(24*x + exp(2*exp(x)) + 4*x^
2 + exp(exp(x))*(4*x + 12) + 36), x)

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sympy [A]  time = 0.14, size = 17, normalized size = 0.77 \begin {gather*} \frac {2 x + 20 e^{x}}{2 x + e^{e^{x}} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)**2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))**2+(4*x+12)*exp(ex
p(x))+4*x**2+24*x+36),x)

[Out]

(2*x + 20*exp(x))/(2*x + exp(exp(x)) + 6)

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